Equation – Atoms


Orbital distances are calculated using the statics rule from classical mechanics that an object will remain at rest when the sum of the forces is zero. This requires an assumption that the proton has an attractive (F1) and repelling force (F2) as described by the pentaquark structure of the proton. In addition, each and every electron in the atom affects one another (F3, F4, F5, etc).

Orbital Location – Sum of Forces is Zero (minimal amplitude)


  • F1: The attractive force is the Coulomb force and is solved between the proton and electron in orbit
  • F2: The repelling force is a repelling axial force between the proton and electron in orbit (see explanation below)
  • F3+: The repelling Coulomb force of each electron in an orbital. One equation is required for each electron in the atom.


The equations for each of the forces (F1, F2, F3+) can be found below in the Explanation section. There is no single equation for orbital distances. All of the forces are simultaneously solved to arrive at the result. The explanation for simultaneous equations is found in the Atomic Orbitals paper, including a simplified approach for solving them via software (e.g. Mathcad). A Mathcad file is available for download for the solutions for hydrogen to calcium.



Explanation of Equation

The axis between the nucleus and the electron being measured is the line where the forces will be calculated (where the sum of the forces is zero).  The attractive force (F1) and proton’s repelling force (F2) are on this axis.  Each additional electron in the atom has a repelling force that may be on a different axis and its distance is computed based on its orbital and the electron angle (see below).


Electron distances in relation to the affected electron (being calculated for orbital distance)


Attractive Force (F1) – Nucleus

The proton in the nucleus has an attractive force based on the number protons (Q1). There is a force on a single electron, so Qwill always be one. This is the Coulomb force. It is the same as the Force Equation, but shown in terms of electron energy (Ee) and radius (re) for simplicity. orbital attractive force f1


Repelling Force (F2) – Nucleus

The proton in the nucleus also has a repelling force based on the number of protons (Q1). Qwill be one again for the single electron being measured. This force is the same as the strong force, but now with a wave passing through two quarks/electrons. The only difference between the strong force equation and this “orbital” equation is that it is now squared after passing through the two quarks/electrons (the strong force passes through one quark/electron before binding with the second). Beyond the standing wave structure of the proton it is a repelling force and diminishes at the cube of distance. The fine structure (αe) is apparent in the equation because of this relationship to the strong force.

orbital repelling force f2

Repelling Force (Fx) – Electrons

The effect from other electrons in the atom are calculated using the same Coulomb force equation as F1 but now the distance is different. The distance (rx) is determined for each electron, requiring multiple equations to be built. It is expanded for each electron in an atom. For example, neon has ten electrons, so this equation is expanded for nine electrons (the tenth electron is the one being calculated).

orbital repelling force from electrons f3


Most electrons in an atom are going to be at different angles than the angle being calculated – the axis between the nucleus and the electron. Therefore, rx is the following:

distance of each electron


Electron Angles (θ)

Every angle is different and needs to be solved. Fortunately, because of tetrahedral alignment from the quarks/electrons in the proton, most angles are at 0° or 60° in relation to the proton.  As orbitals become more complex, they are computed as an average of these angles.  The angle is in relation to the proton because the sum of forces that is calculated in the equations is on the axis between the atom’s nucleus and the affected electron.

Electron Angles

Electron angles in relation to the proton


The s orbitals have a common angle:

s orbital angle


The p orbitals are a mix of these two angles*:

p orbital angle


There are a few exceptions that have this angle**:

exceptions angle


* The angles are averaged across the entire solution
** When sodium and magnesium begin building the 3s orbital, they have this angle


Calculations and Examples

A summary of calculations and examples using these equation rules are provided here. The Mathcad file is here. The remainder of the calculations and examples are detailed in the Atomic Orbitals paper.